An optical lattice is the periodic potential that atoms experience via the ac-Stark shift when they are illuminated by counter-propagating laser beams that form a standing-wave pattern. Optical lattices have been widely used as a versatile platform in cooling, trapping, controlling atoms, and for the study of a variety of problems in physics. The many-body states of ultracold atoms in optical lattices can be characterized by the quantum correlations encoded in time-of-flight images. In this thesis, we mainly discuss the use of the correlations function as a natural framework for characterizing quantum states in optical lattices.

The outline of the thesis is as follows. Chapter 1 gives a brief introduction to optical lattice potentials and ways for describing particles moving in a periodic potential. Chapter 2 explains the importance of correlations and discusses common methods to detect them in cold atom experiments.

Chapter 3 presents work done to study the many-body Schrodinger equation in a quasiperiodic potential and discusses its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We posed a possible visualization of such a connection in experimentally accessible many-body observables. These observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson, and band-insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil's staircases. Our numerical treatment is complemented with a perturbative analysis that provides insight on the underlying physics. The perturbative-theory approach is particularly useful in illuminating the distinction between the Anderson-insulator and the band-insulator phases in terms of paired sets of dimerized states.

Chapter 4 discusses several theoretical procedures developed to understand a recent experiment on macroscopic quantum self-trapping (ST) performed in a 2D optical lattice. Mean field and truncated-Wigner-approximation (TWA) calculations are performed trying to reproduce the experimental observations. The discrepancy between the theory and the experiment lead to the hypothesis of a new type of ST caused by strong correlations. We analyze toy models to support it.

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